General linear methods for ordinary differential equations / Zdzisław Jackiewicz.

Includes bibliographical references (pages 461-476) and index.

Cover -- CONTENTS -- Preface -- 1 Differential Equations and Systems -- 1.1 The initial value problem -- 1.2 Examples of differential equations and systems -- 1.3 Existence and uniqueness of solutions -- 1.4 Continuous dependence on initial values and the right-hand side -- 1.5 Derivatives with respect to parameters and initial values -- 1.6 Stability theory -- 1.7 Stiff differential equations and systems -- 1.8 Examples of stiff differential equations and systems -- 2 Introduction to General Linear Methods -- 2.1 Representation of general linear methods -- 2.2 Preconsistency, consistency, stage-consistency, and zero-stability -- 2.3 Convergence -- 2.4 Order and stage order conditions -- 2.5 Local discretization error of methods of high stage order -- 2.6 Linear stability theory of general linear methods -- 2.7 Types of general linear methods -- 2.8 Illustrative examples of general linear methods -- 2.8.1 Type l: p = r = s = 2 and q = lor 2 -- 2.8.2 Type 2: p = r = s = 2 and q = 1 or 2 -- 2.8.3 Type 3: p = r = s = 2 and q = 1 or 2 -- 2.8.4 Type 4:p = r = s = 2 and q = 1 or 2 -- 2.9 Algebraic stability of general linear methods -- 2.10 Underlying one-step method -- 2.11 Starting procedures -- 2.12 Codes based on general linear methods -- 3 Diagonally Implicit Multistage Integration Methods -- 3.1 Representation of DIMSIMs -- 3.2 Representation formulas for the coefficient matrix B -- 3.3 A transformation for the analysis of DIMSIMs -- 3.4 Construction of DIMSIMs of type 1 -- 3.5 Construction of DIMSIMs of type 2 -- 3.6 Construction of DIMSIMs of type 3 -- 3.7 Construction of DIMSIMs of type 4 -- 3.8 Fourier series approach to the construction of DIMSIMs of high order -- 3.9 Least-squares minimization -- 3.10 Examples of DIMSIMs of types 1 and 2 -- 3.11 Nordsieck representation of DIMSIMs -- 3.12 Representation formulas for coefficient matrices P and G183; -- 3.13 Examples of DIMSIMs in Nordsieck form -- 3.14 Regularity properties of DIMSIMs -- 4 Implementation of DIMSIMs -- 4.1 Variable step size formulation of DIMSIMs -- 4.2 Local error estimation -- 4.3 Local error estimation for large step sizes -- 4.4 Construction of continuous interpolants -- 4.5 Step size and order changing strategy -- 4.6 Updating the vector of external approximations -- 4.7 Step-control stability of DIMSIMs -- 4.8 Simplified Newton iterations for implicit methods -- 4.9 Numerical experiments with type 1 DIMSIMs -- 4.10 Numerical experiments with type 2 DIMSIMs -- 5 Two-Step Runge-Kutta Methods -- 5.1 Representation of two-step Runge-Kutta methods -- 5.2 Order conditions for TSRK methods -- 5.3 Derivation of order conditions up to order 6 -- 5.4 Analysis of TSRK methods with one stage -- 5.4.1 Explicit TSRK methods: s = l, p = 2 or 3 -- 5.4.2 Implicit TSRK methods: s = l, p = 2 or 3 -- 5.5 Analysis of TSRK methods with two stages -- 5.5.1 Explicit TSRK methods: s = 2, p = 2, q = 1 or 2 -- 5.5.2 Implicit TSRK methods: s = 2, p = 2, q = 1 or 2 -- 5.5.3 Explicit TSRK methods: s = 2, p = 4 or 5 -- 5.5.4 Implicit TSRK methods: s = 2, p = 4 or 5 -- 5.6 Analysis of TSRK methods with three stages -- 5.6.1 Explicit TSRK methods: s = 3, p = 3, q = 2 or 3 -- 5.6.2 Implicit TSRK methods: s = 3, p = 3, q = 2 or 3 -- 5.7 Two-step collocation methods -- 5.8 Linear stability analysis of two-step collocation methods -- 5.9 Two-step collocation methods with one stage -- 5.10 Two-step c.

Learn to develop numerical methods for ordinary differential equations General Linear Methods for Ordinary Differential Equations fills a gap in the existing literature by presenting a comprehensive and up-to-date collection of recent advances and.

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